NOTE: Here I must use the " * " for the dot, the " > " for the horseshoe, and the " _ " for the triple bar.
Statement variables (p, q, r, É) are used to construct statement forms. Thus, the statement form p v q could be used to stand for any disjunction:
A v B
(A * B) v [C É > (D _ E)]
~ C v (D * ~E)
(1) You must be able to replace each variable with the same statement
name throughout.
A v (B * A) is an instance of p v (q * p) but A v (B * C) is not.
Part of what the form reveals is that the content of the first disjunct
is the same as the content of the second conjunct, whatever that content
might happen to be.
This doesn't rule out the possibility that q could stand for the same
content. (See next slide)
(2) Different variables can take the same statement name.
A v (A * A) and B v (B * B) are both instances of p v (q * p) as is A v (B * A)
q can stand for what p stands for, but no single variable can stand for different things in the same statement.
A v (A * B) is not an instance since 'p' cannot stand for both 'A' and
'B' in the same statement.
(3) Variables can stand for simple or compound statements
Which of the following are instances of (p v q) É > p? (Read: A conditional whose antecedent is a disjunction)
(~ A v B) É > (B * C)
~ (A v B) É > B
(B v B) É > B
[(B C) v B] É > (B * C)
(~ A v B) É > (B * C)
NO. The replacement for p changes.
~ (A v B) É > B
NO. The antecedent is a negated disjunction, not a disjunction.
(B v B) É > B
YES. It fits the form. (Both p and q can be replaced with the same content.)
[(B C) v B] É > (B * C)
YES. It fits the form. (p is replaced with the compound statement (B
* C).)
The truth value of a compound statement depends upon the truth values of the component parts.
Statements that aren't truth functional
"Lucy believes that Ricky is at the club" is a compound statement
whose truth value does not depend on the truth value of "Ricky is
at the club."
Truth tables are an organized arrangement of truth values showing every
possible combination of truth value assignments for the simple and compound
statements involved.
The array of truth values is always built the same way to limit error
and to guarantee that all possible combinations of truth value assignments
are accounted for.
Truth tables can be used to define logical operators and connectives.
Negation
Conjunction
Disjunction
Material Conditional
Bi-conditional
If Lucy wears the decoder ring and Ethel knows the password, then Ricky
suspects there is a spy.
(R * P) É > S
where Lucy wears the ring, Ethel doesn't know the password and Ricky
suspects nothing.
(T * F) É > F
To find the truth of the compound proposition determine first the truth value of the conjunction (the minor connective) and then determine the truth value of the conditional (the major connective).
(T * F) É > F
(F) > ÉF
T
~ A v ~ ( A * ~ B )
Suppose A is true and B is false.
This is a disjunction so the " v " will be the last value determined.
The propositions left and right of the "v" must be determined
separately.
First the left side
~ A v ~ ( A * ~ B )
~ T v ~ ( T * ~ F )
F v ~ ( A * ~ B )
The right side is more complicated
Begin inside the parentheses. You must determine the value of the negated
"B" before the determine the value of the conjunction. And you
must determine the value of the conjunction before you determine the value
of the negated conjunction.
F v ~ ( T * ~ F )
F v ~ ( T * T)
F v ~ ( T )
F v F
F
You will fail the course unless you attend class every week.
F v A
Notice that you could still attend every week and fail. What's ruled out is that you not attend every week and not fail (~ F A).
This is sometims called the inclusive sense of "unless".
Ordinary language and "unless"
You will pass the exam unless you are lazy.
P v L
Here both can't be false (~ P * ~ L) and you can't both pass the exam
and be lazy (P * L).
This is sometimes called the exclusive sense of "unless".
The material condition expresses a relationship based only on the truth
values of the consequent and antecedent.
In ordinary language we often use conditionals to express more significant
relationships.
Thus puzzles can arise ....
p Éq is true whenever the antecedent
if false.
If Michael Jackson is a whale, then he is a reptile.
If Clinton advocates drug abuse, then he is a good President.
These conditionals are true because they have false antecedents. If
you think they are false, perhaps you are thinking of a different sort
of conditional ....
If Michael Jackson were a whale, then he would be a reptile.
If Clinton were to advocate drug abuse, then he would be a good President.
Here we are suggesting that there is more to the relationship between
antecedent and consequent than the fact that the latter is not false when
the former is true.